SEMINAR

Phase Space and Path Integral Methods in Multidimensional Wave Propagation

Naval Research Laboratory
Stennis Space Center

ABSTRACT

The n-dimensional, elliptic, two-way, Helmholtz wave propagation problem can be exactly reformulated in a well-posed manner as a one-way wave propagation problem in terms of appropriate square-root Helmholtz and Dirichlet-to-Neumann (DtN) operators. This reformulation has application for (1) large-scale, deterministic, direct wave propagation modeling, (2) the development of layer-stripping algorithms for the corresponding, multidimensional, ill-posed inverse problems, and (3) the formulation and computation of stochastic wave propagation models. These operators are formally defined and constructed in an appropriate pseudodifferential operator calculus in terms of their corresponding operator symbols. The fundamental wave equation solutions (propagators) are expressed as path integrals, directly in terms of the operator symbols, which result in explicit, marching (one-way) computational algorithms. The analysis and computation of both direct and inverse wave propagation problems can then be largely reduced to the understanding and exploitation of the operator symbol (singularity and oscillatory) structure, and the subsequent construction of uniform (over phase space) asymptotic operator symbol approximations. Since the relevant (frequency-domain) operators lie outside of the well-developed theory and asymptotic analysis of elliptic pseudodifferential operators, new, uniform, high-frequency asymptotic operator symbol expansions are developed. This analysis uniformly incorporates the usual algebraic terms associated with the elliptic calculus and the terms of exponential order, corresponding to the contributions from the infinitely-smooth part of the kernel, lying outside of the standard theory. Both low-frequency and multiscale, high-frequency asymptotic results are also obtained. These asymptotic results are compared with several exact operator symbol constructions incorporating canonical features and including (1) both the focusing and defocusing quadratic profiles, (2) a class of hyperbolic function profiles incorporating symmetric and asymmetric wells and large gradient effects, (3) the delta function profile, and (4) piecewise-constant profiles such as the discontinuity and rectangular well. Examples from both direct and inverse scattering will be given.

WHERE: TEC 205

WHEN(day): Friday, November 17th, 2000

WHEN(time): 2:00pm

EVERYBODY IS INVITED